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I want to make 1,000 picks (without replacement) from sample of 10,000,000 (1,000 types and 10,000 of each type)

What is the probability that x% of all types are picked from my 1,000 picks?

Is there a formula / distribution I can use in excel to calculate this?

mikec27
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1 Answers1

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The number selected of a given type will be well represented by a Poisson distribution with mean $1$ because on average you are selecting $1$ of each type. In particular, about $\frac 1e$ of the types will not be selected. A good approximation for the probability a particular number of types are chosen is to ignore the correlation between the types. You can then make a binomial distribution with probability $\frac 1e$ a type is not selected and probability $1-\frac 1e$ that it is. Excel has functions for each of the distributions, but the numbers will be too large for the binomial. You can use the normal approximation, where the standard deviation is $\sqrt{10,000 \frac {1-e}{e^2}}$ and use the error function to find out if the number is in a given range.

Ross Millikan
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  • That’s great Ross thanks – mikec27 Jul 13 '20 at 14:23
  • Do you know how I would do that in excel? – mikec27 Jul 13 '20 at 14:23
  • For example, $34%$ of the data lies between the mean and mean $+1\sigma$, so between $\frac {10,000}e$ and $\frac {10,000}e+\sqrt{10,000 \frac {1-e}{e^2}}$. You can look up the cumulative error function to get the fraction in any interval you like. – Ross Millikan Jul 13 '20 at 14:35